Method and arrangement for predicting measurement data by means of given measurement

ABSTRACT

A method and an arrangement are provided for predicting measurement data using given measurement data, in which a stochastic process is matched to the given measurement data. Simulation runs are carried out from a given time-point until a final time-point. The forecast measurement data is determined for each simulation run. Measurement data for the final time-point is predicted within a range of values, which is determined by the forecast measurement data.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method and arrangement for predictingmeasurement data using given measurement data.

2. Description of the Related Art

A technical system often requires facilities for forecasting based onknown (measurement) data, particularly in the context of errorsusceptibility or cost estimates.

Forecasts generated by experts are generally subject to errors. Expertscannot carry out exact analyses, at least of highly complex systems.

A stochastic point process, in particular a Poisson process, isdescribed in Sidney I. Resnick: “Adventures in Stochastic Processes”,Birkhtäuser Boston, 1992, ISBN 3-7643-3591-2, pp. 303-317 (Resnick).

SUMMARY OF THE INVENTION

The object of the invention is to allow the automatic prediction(forecast) of measurement data using given measurement data.

This object is achieved in accordance with the method and apparatusdescribed below; developments of the invention are also described in thefollowing text.

In order to achieve this object, a method is provided for predictingmeasurement data using given measurement data, in which a stochasticprocess is matched to the given measurement data. Simulation runs arecarried out from a given time-point until a final time-point. Theforecast measurement data is determined for each simulation run.Measurement data for the final time-point is predicted within a range ofvalues, which is governed by the forecast measurement data.

One development is to define a confidence range for the prediction ofmeasurement data, where the a % lowest and b % highest forecastmeasurement data are eliminated. In particular, a % can equal b %. Forexample, a 95% confidence range can thus be defined by ignoring the 2.5%lowest and 2.5% highest forecast measurement data.

One advantage is that the measurement data can be predicted (forecast)with an accuracy that is within a confidence range, from a giventime-point. This makes it possible to identify, e.g., the feasibility orimpossibility of a task associated with the measurement data, at anearly stage. Appropriate measures can therefore be initiated in order tocounteract forecast impossibility.

This is particularly important in the case of a complex system, e.g., asoftware development process, where the extent to which a schedule canbe followed before the software is completed can be shown in asubsequent test phase. Even more important in this context is theability to adopt countermeasures at an early stage if a delay has beenclearly identified, e.g., in an integration test phase. This firstlyaffects the feasibility of the specified deadline (timescale) andsecondly directly affects costs, since non-compliance with the agreedtimescale often results in additional costs.

One refinement is for the stochastic process to be a non-homogeneousPoisson process.

In particular, the measurement data may in one refinement comprisenumbers of errors. This applies to software development, for example,where the level of maturity is documented in accordance with the errorsmeasured in a test phase. Completion is directly dependent on this levelof maturity. In other words, the software cannot be delivered tocustomers until most of the errors have been removed from the software.This is particularly important with regard to resources (required totest and correct errors) and costs (due to delayed delivery).

In order to achieve the object of the invention, a method is alsoprovided for predicting measurement data using given measurement data,in which a stochastic process is matched to the given measurement data.A range is ascertained, by sorting the probability values generated bythe stochastic process according to size, around an expected value.Measurement data is predicted on the basis of this range, and inparticular the probability values within the range.

One development is for the probability values generated by thestochastic process to be sorted symmetrically by size around theexpected value. In particular, this means that the highest probabilityvalue represents the middle of the range, i.e., the expected value,whereas the next highest probability value is arranged to the right orleft of the expected value. The next highest probability value is thenarranged symmetrically on the other side of the expected value, in turn.

This analytical (design) procedure provides a range, where the breadthof the range in turn indicates which probability values are significantin the prediction of the measurement data.

In one particular refinement, the breadth of the range is determined byignoring the probability values that lie below a given threshold.

This produces a range (confidence range), which has a specific breadthas a result of the threshold. This breadth corresponds to the certaintywith which the measurement data is predicted.

If one assumes that the stochastic process is a non-homogeneous Poissonprocess, then the non-homogeneous Poisson process defines a step size,particularly on a time axis t, which indicates when the next error willoccur. One characteristic of the non-homogeneous Poisson process is thatit has no memory, so that a “no-memory” search is carried out from eacherror that occurs at a specific time-point, for a time-point thatindicates the next error.

In order to achieve the object of the invention, an arrangement is alsoprovided for predicting measurement data using given measurement datathat has a processor unit and is configured in such a way that:

-   -   a) a stochastic process can be matched to the given measurement        data;    -   b) simulation runs can be carried out from a given time-point        until a final time-point;    -   c) the forecast measurement data can be determined for each        simulation run; and    -   d) the prediction of measurement data for the final time-point        can be predicted within a range of values, which is determined        by the forecast measurement data.

In order to achieve the object of the invention, an arrangement isfurther provided for predicting measurement data using given measurementdata that has a processor unit and is configured in such a way that:

-   -   a) a stochastic process can be matched to the given measurement        data;    -   b) a range can be ascertained by sorting probability values        generated by the stochastic process according to size around an        expected value; and    -   c) the measurement data is predicted within the limits of the        range.

The arrangements are particularly suitable for carrying out theinventive method or the developments described above.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments of the invention are shown and explained belowwith reference to the drawings, in which:

FIG. 1 is a graph showing an accumulated number of errors over a testperiod;

FIG. 2 is a graph showing the superimposed confidence ranges fordifferent process models;

FIG. 3 is a flowchart showing the steps in a method for predictingmeasurement data using given measurement data;

FIG. 4 is a further flowchart showing the steps in a method forpredicting measurement data using given measurement data; and

FIG. 5 is a block diagram showing a processor unit.

DETAILED DESCRIPTION OF THE INVENTION

In order to be able to forecast a number of expected errors in atechnical process, e.g., in a software development process,non-homogeneous Poisson processes (NHPP) are calibrated (i.e., matchedto measurement data, such as the occurrence of errors over time) asfollows:

The following equation describes a counting process associated with thestochastic point process (non-homogeneous Poisson process):{N(t)}_(tεR) ₊   (1)

-   -   and a time-point t₀ defines the end of a test period, i.e., a        time-point at which the given data ends. The stochastic        processes        {U(t)}_(tεR) ₊ and  (2)        {O(t)}_(tεR) ₊   (3)    -   are searched with        p(U(t)≦N(t)−N(t ₀)≦o(t)|N(t ₀)=n ₀)≧α  (4)    -   for all time-points where t>t0 and given values αε (0.1)        (confidence level) and n0 ε N. In particular, the following text        examines the increases in the stochastic countings process in        relation to the time-point t₀.

In the present case, where equation (1) represents a non-homogeneousPoisson process, the following equation (cf. Resnick) $\begin{matrix}{{P\left( {{{N\left( t_{1} \right)} - {N\left( t_{0} \right)}} = l} \right)} = {{\exp\left( {- \left\lbrack {{i\left( t_{1} \right)} - {i\left( t_{0} \right)}} \right\rbrack} \right)} \cdot \frac{\left\lbrack {{i\left( t_{1} \right)} - {i\left( t_{0} \right)}} \right\rbrack^{l}}{l!}}} & (5)\end{matrix}$

-   -   applies for        0≦t ₀ <t ₁ >∞, l εN ₀  (6)    -   and an intensity (mean measure, mean value function) of        i: R ⁺ →R ⁺ ,t|→i(t)=EN(t)  (7).

Since the nature of the Poisson process dictates that the increases(error increases in this case) are independent of previous increases,equation (5) for the time-points t>t₀ to define a (minimum) range

 [g _(u) , g _(o) ]=[g _(u)(t), g _(o)(t)]⊂N ₀  (8)

-   -   can be simplified to $\begin{matrix}        {{\sum\limits_{l = g_{u}}^{g_{o}}{p\left( {{{N(t)} - {N\left( t_{0} \right)}} = l} \right)}} \geq {\alpha.}} & (9)        \end{matrix}$

Due to the unimodal nature of the Poisson count density, a range [g_(u),g_(o)] can be determined as follows:

Step 1: Sort the elementary probabilitiespl:=P(N(t)−N(t ₀)=l), l εN ₀

-   -   into descending order and label the values sorted thus using    -   p₍₀₎, p_((i)), . . . (i.e. {p₀p₁, . . . }={p₍₀₎, p_((i)), . . .        } and        -   p₍₀₎, ≧p₍₁₎≧. . . );

Step 2:${{Determine}\quad l_{\min}}:={\min\left\{ {{l \in {N_{0}\left. {{\sum\limits_{i = 0}^{l}{P(i)}} \geq \alpha} \right\}}};} \right.}$

Step 3: Determine an index set

-   -   I:={i₀, . . . , i_(l) _(min) }⊂N₀ where    -   {pi₀, . . . , pi_(l) _(min) }={p₍₀₎, . . . , p_((l) _(min)) };

Step 4: Substitute$g_{u}:={{\min\limits_{i \in I}{\left\{ i \right\}\quad{and}\quad g_{o}}}:={\max\limits_{i \in I}{\left\{ i \right\}.}}}$

The range from equation (8) is also referred to as the forecast range.

Stochastic Simulation (second approach)

It is possible to determine the confidence range described usingsimulation, with the following steps:

Step 1: Start independent simulation runs based on the selected processmodel at time-point t0 of the last error message m ε N;

Step 2: End a simulation run as soon as the required final time-pointt_(e) is reached;

Step 3: Repeat Step 2 until all simulation runs are finished;

Step 4: Sort the numbers {circumflex over (N)}_(i)(t_(e)) of the errorsgenerated in the i-th simulation run in the time period (t₀, t_(e)),i=1, . . . , m, in descending order, and label the values sorted thus{circumflex over (N)}₍₁₎(t_(e)), . . . , {circumflex over(N)}_((m))(t_(e)); and

Step 5: Substitute

-   -   ĝ_(u):={circumflex over (N)}_((└m·α/2┘))(t_(e)) and    -   ĝ_(u):={circumflex over (N)}_((└m·(1−α/2)┐))(t_(e)),

i.e., eliminate the (100·(1−α)/2) % lowest and highest values.

This produces the confidence range directly.

Each individual simulation run is based on a simulation algorithm, whichis known from (cf. Brately, et al., 1987):

The simulated generation of intermediate arrival times for anon-homogeneous Poisson process is as follows:

Step 1: Substitute${\overset{\_}{\lambda}:={\sup\limits_{t \geq t_{s}}\left\{ {\lambda(t)} \right\}}},$$\begin{matrix}{{{{\lambda(t)}:=\frac{\mathbb{d}i}{\mathbb{d}t}}}_{t}.} & (10)\end{matrix}$

Step 2: Generate a (pseudo) random variable X that is exponentiallydistributed with the parameter {overscore (λ)}, i.e., x:=−log(U)/{overscore (λ)}, where U is equally distributed over (0,1);

Step 3: Generate a random variable U that is equally distributed over(0,1); and

Step 4: If U≦λ(t_(s)+x)/{overscore (λ)}, then substitute t=t_(s)+X;otherwise substitute t_(s)=t_(s)+X and go to Step 1.

The example graph in FIG. 1 shows an accumulated number of errors duringa given test period. From time-point t₀, it shows a prediction range Klfor all time-points t₀+x.

The intensity i is normally derived from equation (10) for λ. Forexample the result is as follows:

a) λ(t)=a·b·c·exp(−bt ^(c))·t^(c−1)

(λ(t) is strictly monotonously descending for c≦1, and unimodal for c>1with a definitive maximum at a point$\left. {t_{\max} = \sqrt[c]{\frac{c - 1}{b\quad c}}} \right).$

b) Otherwise, {overscore (λ)} is derived in accordance with the above iscomments as follows: $\overset{\_}{\lambda} = \left\{ {\begin{matrix}{{\lambda\left( t_{s} \right)},} & {\left( {c \leq 1} \right) ⩔ \left( {t_{s} \geq t_{\max}} \right)} \\{\lambda\left( t_{\max} \right)} & \quad\end{matrix}.} \right.$

The graph in FIG. 2 shows the superimposed confidence ranges. Inparticular, this illustrates that possible forecasts become morescattered the further they extend into the future. In particular,confidence ranges calculated using different process models can bedemonstrated in the same way as shown in FIG. 2.

FIG. 3 shows a flowchart for the steps of a method for predictingmeasurement data using given measurement data. In Step 301, a stochasticprocess, in particular a non-homogenous Poisson process (to represent astochastic count process), is matched to given measurement data. In Step302, simulation runs are run from time-point t₀ to a final time-pointt_(e) that is to be forecast. In Step 303, for each simulation run,forecast measurement data is determined and a prediction of measurementdata is restricted to a range which is covered by the measurement datadetermined by the simulation runs (see Step 304). In Step 305, aconfidence range is determined in which a given proportion of the lowestand highest forecast measurement data is ignored in each case (thiscorresponds to the aforementioned range). The method terminates in Step306.

FIG. 4 shows a further flowchart for the steps of a method forpredicting measurement data using given measurement data. In Step 401, astochastic process, in particular a non-homogenous Poisson process, ismatched to the given measurement data. Probability values are determinedusing the stochastic process, and these are sorted according to sizearound an expected value (see Step 402). This sort operation results inthe definition of a range, namely a confidence range in this case. Thebreadth of the confidence range is determined by comparing theaccumulated probabilities with a given threshold. As described above,the confidence range gives a distribution or uncertainty, respectively,of a time-point to in the future, which allows the measurement data tobe estimated in the future (see Step 403). The method terminates in Step404.

FIG. 5 shows a processor unit PRZE that may be used to implement theinventive method. The processor unit PRZE comprises a processor CPU, amemory unit MEM, and an input/output interface IOS, which is used indifferent ways via an interface IFC: a graphics interface allows outputto be viewed on a monitor MON and/or output to a printer PRT. Inputs areentered via a mouse MAS or a keyboard TAST. The processor unit PRZE alsoincludes a data bus BUS, which provides the connection between a memoryunit MEM, the processor CPU and the input/output interface IOS. It isalso possible to connect additional components to the data bus BUS, e.g.additional memory, data storage (hard disk) or scanner.

The C programming language is used in the following examples, which showan algorithm to define confidence ranges for forecasts and an algorithmfor simulated definition of confidence ranges for forecasts.

Program 1:

-   -   /* Definition of confidence ranges for forecasts */    -   /* based on the generalized Goel-Okomoto model */

#include <stdlib.h> #include <math.h> #include <stdio.h> #define true 1#define false −1 double mv_genGO(double,double,double,double): doublepoisson(double,long): void ki_nhpp( ): int main(argc,argv) int argc;char *argv[]; { double a,b,c,bt,st,kn; long low,upp,lauf; if (argc<7) {print(“\n\nZuwenig Argumente! \n\n”); print(“Aufruf: %s <Par1> <Par2><Par3> <Startzeit> <Endzeit>”, “<KNiveau>\n\n”, argv[0]); return 1;  } a= atof(argv[1]); b = atof(avgv[2]); c = atof(argv[3]); bt=atof(argv[4]);st= atof(argv[5]); kn= atof(argv[6]); for (lauf=1;lauf< ;lauf++) {ki_nhpp(mv_genGO,a,b,c,bt,bt+lauf*(st-bt)/10.,kn,&low,&upp);printf(“Zertpunkt; %8.2f Fehlerintervall: [%d,%d]\n”,bt+lauf*(st-bt/10., low, upp);  } return 0: } double mv_genGO(x,a,b,c)double x,a,b,c; { return( a*(1.0-exp(-b*pow(x,c))) ); ) doublepoisson(lambda,wert) double lambda; long wert; { long i; doubleitval,hv; if (lambda<600) { itval = exp(-lambda): for (i=wert;i>=1;i−−){ itval *= lambda/(double)i; }  }

else { hv = exp(-lambda/(double)wert); itval = 1.0; for(i=wert;i>=1;i−−) { itval *= lambda/(double)i*hv; }  } return ( itval );} void ki_nhpp(mv_nhpp, par1_nhpp, par2_nhpp, par3_nhpp, start_time,stop_time, k_niveau, lower, upper) doublemv_nhpp(double,double,double,double);   double par1_nhpp, par2_nhpp,par3_nhpp, start_time, stop_time, k_niveau; long *lower, *upper; { longlauf; int lborder,mod_low,mod_upp; double sum,tmp_mv, val_l, val_u;tmp_mv = mv_nhpp(stop_time,par1_nhpp.par2_nhpp,par3_nhpp) -mv_nhpp(start_time,par1_nhpp,par2_nhpp,par3_nhpp); lauf = (long)tmp_mv;*lower = lauf−1; *upper = lauf+1; mod_low= false; mod_upp= false; sum =poisson(tmp_mv,lauf); val_l =   poisson(tmp_mv, *lower); val_u =poisson(tmp_mv,*upper); while (sum<k_niveau) { if (val_i<val_u) { sum +=val_u; (*upper)++; lborder = false; mod_upp = true; val_u =poisson(tmp_mv,*upper);   } else { sum += val_l; (*lower)−−; lborder =true; mod_low = true; val_l = poisson(tmp_mv,*lower);   }  } if (lborder== true) { (*lower)++; } else   {(*upper)−−;) if (mod_low == false) {(*lower)++; } if (mod_upp == false) { (* upper)−−; } return: }Program 2:

-   -   /* Simulated definition of confidence ranges for forecasts */    -   /* based on the generalized Goel-Okomoto model */

#include <stdlib.h> #include <math.h> #include <time.h> #include<stdio.h> #include <values.h> #define true 1 #define false −1 doubledrand48(void); void srand48(long); double sim_exp(double); doublelambda_genGO(double,double,double,double); void sim_nhpp( ); intmain(argc,argv) int argc; char *argv[]; { time_t t; doublea,b,c,bt,st,pnt[1000000],check_time[12]; long lauf,no_pnt,seed_run; intclauf; FILE *datei: if (argc<6) { printf(“\n\nZuwenig Argumente! \n\n”);printf(“Aufruf. %s <Par1> <Par2> <Par3> <Startzeit> <Endzeit>\n\n”,argv[0]; return 1;  } datei = fopen(“sim.seed”,“r”); if (datei==NULL) {seed_run = 1;  } else { fscanf(datei,“%8”, &seed_run); fclose(datei);seed_run++;  } datei = fopen(“sim.seed”,“w+”); fprintf(dtei, “%6d\n”,seed_run ); fclose(datei); time (&t); /* initialisierung des */ t +=seed_run*100; /* Zufallszahlengenerators */ srand48 ((unsigned long) t); /* mit Hilfe der Systemzeit */ a = atof(argv[1]); b = atof(argv[2]); c= atof(argv[3]): bt= atof(argv[4]): st= atof(argv[5]);sim_nhpp(lambda_genGO,a,b,c,bt,st&pnt,&no_int); for(lauf=1;lauf<=no_pnt;lauf++) { printf(“%15.7f % 10d \n”, pnt[lauf],lauf);  }

datei = fopen(“ki.tmp”,“a”); for (lauf=1;lauf< ;lauf++) {check_time[lauf] = bt+lauf*(st-bt)/10.;   } check_time[11] =pnt(no_pnt]+1; /* groBer als die groBte simulierte Zeit */ clauf = 1;for (lauf= 1;lauf<=no_pnt;lauf++) { while(pnt[lauf])>=check_time[clauf]) { fprint(datei, “%8.2f %6d ”,check_time[clauf], lauf−1); clauf++;   }  } if (pnt[no_pnt]<check_time[10]) { for (lauf=clauf,lauf< ;lauf++) { fprint(datei, “%8.2f%6d ”, check_time[lauf], no_pnt);   }  } fprintf(datei. “\n”);fclose(datei); return 0; } double sim_exp(lambda) double lambda;{return( -log(dand48( ))/lambda ); } double lambda_genGO(x,a,b,c) doublex,a,b,c; { return( a*b*c*pow(x,c-1)*exp(-b*pow(x,c)) ); } voidsim_nhpp(lamba_nhpp, par1_nhpp, par2_nhpp, par3_nhpp, start_time,stop_time, path, no_points) doublelambda_nhpp(double,double,double,double); double   par1_nhpp, par2_nhpp,par3_nhpp, start_time, stop_time; double path[]; long *no_points; {double sim_time,x,u,x_bar,lambda_bar; *no_points=0; sim_time =start_time; do ( if (par3_nhpp<=1) { lambda_bar =lambda_nhpp(sim_time,par1_nhpp,par2_nhpp,par3_nhpp);   } else { x_bar =pow((par3_nhpp-1.0/par2_nhpp/par3_nhpp,1.0/par3_nhpp); if(sim_time>=x_bar) { lambda_bar =lambda_nhpp(x_bar,par1_nhpp,par2_nhpp,par3_nhpp);    } else { lambda_bar= lambda_nhpp(sim_time,par1_nhpp,par2_nhpp,par3_nhpp);    }   } x =sim_exp(lambda_bar); u = drand48( );  if(u<=lambda_nhpp(sim_time+x,par1_nhpp,par2_nhpp,par3_nhpp)/lambda_bar){(*no_points)++;     path[*no_points]=sim_time+x;    }  sim_time+=x;   } while (sim_time<=stop_time);  return;  }Program 3:

-   -   /*Definition of confidence ranges from the simulation data */    -   /*(the simulation data is sorted into ascending order) */

#include <stdlib.h> #include <math.h> #include <stdio.h> intqsort_icmp(int*,int*); int qsort_icmp(x,y) int *x, *y; { if (*x<*y)  {return ( −1 ); ) else if (*x==*y) { return ( 0 ); } else   { return( 1); } } int main(argc,argv) int argc; char *argv[]; { intpnt[11][100000]; int qs[100000]; char *dname; int frac,i; longlauf,lower_bound,upper_bound; long l,no_pnt,seed_run; doublectime[11],x; FILE *datei; if (argc<3) { printf(“\n\nZuwenig Argumente!\n\n”); print(“Aufruf: %s <Dateiname> <Konfidenzniveau (in    %%)>\n\n”, arg[0]); return 1;  } dname = argv[1]; frac =100-atoi(argv[2]); lauf = 0; datei = fopen(dname,“r”); if (datei==NULL){ return 1; } else { while (!feof(datei)) { lauf++; for (i=1;i<=9,i++) {fscanf(datei,“%8lf %6d ”, &ctime[i], &pnt[i][lauf]);     }fscanf(datei,“%8lf %6d \n”, &ctime[10], &pnt[10][lauf]);    }fclose(datei);   } lower_bound = (long)floor(lauf*frac/200.);upper_bound = (longceil(lauf*(200.-frac)/200.); if (lower_bound<1){lower_bound = 1;}

printf(“\n\n%2d%%-Sicherheitsbereich bei %d Simulationslaufen\n\n”,100-frac,lauf); for (i=1;i< ;i++) { for (l= 1;l<=lauf,l++) { qs[l] =pnt[l][l];   } qsort(&qs[1], lauf, sizeof(int), &qsort_icmp);printf(“Zeitpunkt; %8.2f Fehlerintervall: [%d,%d]\n”, ctime[i],qs[lower_bound], qs[upper_bound]);  } return 0; }

The above-described method and apparatus are illustrative of theprinciples of the present invention. Numerous modifications andadaptations will be readily apparent to those skilled in this artwithout departing from the spirit and scope of the present invention.

1. A method for predicting measurement data until a final time-pointusing given measurement data, comprising: matching, using a processor, astochastic process to said given measurement data; running simulationruns of said stochastic process from a given time-point until said finaltime-point; determining forecast measurement data for each simulationrun; predicting measurement data by stating a range of values, which isdetermined by said forecast measurement data, and providing saidpredicted measurement data as useable output; determining a confidencerange for said prediction of measurement data; and eliminating a lowestpercentage and a highest percentage forecast measurement data.
 2. Themethod as claimed in claim 1, wherein the lowest and highest percentagesare equal values.
 3. The method as claimed in claim 1, wherein saidstochastic process is a non-homogeneous Poisson process.
 4. The methodas claimed in claim 1, wherein said measurement data represents numbersof errors.
 5. A method for predicting measurement data using givenmeasurement data, comprising: matching, using a processor, a stochasticprocess to said given measurement data; sorting probability valuesgenerated by said stochastic process according to size, to provide arange around an expected value; and predicting measurement data withinlimits of said range, and providing said predicted measurement data asuseable output; and determining a confidence range for said predictionof measurement data; and eliminating a lowest percentage and a highestpercentage forecast measurement data.
 6. The method as claimed in claim5, further comprising: sorting said probability values generated by saidstochastic process symmetrically by size around said expected value. 7.An arrangement for predicting measurement data until a final time-pointusing given measurement data, comprising: a processor unit, having aCPU, bus, memory, and input/output controller, configured in such a waythat: simulation runs of the stochastic process can be carried out froma give time-point until the final time-point; forecast measurement datacan be determined for each simulation run; and measurement data ispredicted by stating a range of values, which is determined by saidforecast measurement data, said measurement data being output in auseable form; and a confidence range for said prediction of measurementdata is determined; and a lowest percentage and a highest percentageforecast measurement data are eliminated.
 8. An arrangement forpredicting measurement data using given measurement data, comprising: aprocessor unit, having a CPU, bus, memory, and input/output controller,configured in such a way that: a stochastic process can be matched tothe given measurement data; a range can be ascertained by sortingprobability values generated by said stochastic process according tosize around an expected value; and said measurement data is predictedwithin the limits of the range, said measurement data being output inuseable form; and a confidence range for said prediction of measurementdata is determined; and a lowest percentage and a highest percentageforecast measurement data are eliminated.